Positive critical Hausdorff measure conjecture for random Cantor sets

Let $F$ be the random Cantor set generated by the tree of coin flips with contraction ratio $r$, let $M>N$, and let the symmetric probability vector be

$$\mathbf{p}=(1/N,\dots,1/N).$$

Set

$$s=-\frac{\log N}{\log r}.$$

Positive critical Hausdorff measure conjecture. With probability one with respect to $\mu_{\mathbf{p}}$,

$$\mathcal{H}^s(F)>0.$$

The preceding dichotomy result shows that the normalized expected number of surviving intervals converges to a positive limit when $M>N$, suggesting positivity of the Hausdorff measure at the critical dimension. The claim is stated as a conjecture because the paper does not establish it.